“Dimension” can describe independent coordinates, the rank of a readout, the growth of a neighbourhood or the scaling of a measurement. These meanings can agree in familiar settings, but they are not interchangeable definitions.
Name the diagnostic
The three-dimensional readout essay studies the rank of a linear map: how many independent real outputs it can produce. A different question asks how the number of accessible sites grows as we increase a distance threshold.
If a neighbourhood contains approximately N(r) = C rᴰ sites over a range of radii, doubling r multiplies the count by approximately 2ᴰ. The diagnostic D = log₂[N(2r)/N(r)] estimates that growth exponent over the chosen range. It is not a universal definition valid for every kind of space.
Count two neighbourhoods
On the infinite integer line, connect each integer to its two immediate neighbours. The graph ball of integer radius r around zero contains N₁(r) = 2r + 1 vertices. On the infinite square lattice with horizontal and vertical edges, the graph ball is a diamond containing N₂(r) = 2r² + 2r + 1 vertices.
The second count follows by adding the shells: at graph distance j there are 4j sites, so the total is 1 + 4(1 + 2 + … + r). Both constructions use minimum edge count as distance.
| Graph | N(10) | N(20) | Doubling exponent |
|---|---|---|---|
| Integer line | 21 | 41 | log₂(41/21) ≈ 0.965 |
| Square lattice | 221 | 841 | log₂(841/221) ≈ 1.928 |
The finite-radius estimates are close to one and two, but not exactly equal to them. Their large-radius limits are one and two because the leading terms grow as r and r². A scaling claim should identify its limiting procedure or the finite interval used to estimate it.
A finite system eventually saturates
Now close a line of 100 vertices into a ring. Before the radius explores much of the ring, neighbourhood growth resembles a line. Once the ball includes all 100 vertices, increasing the radius adds nothing. The doubling diagnostic then gives log₂(100/100) = 0.
That zero describes saturation of this count in a finite system. It does not show that a macroscopic circle has lost its local one-dimensional character. The measurement range has reached the size of the whole system.
At the opposite end, radii smaller than the graph's edge scale also fail to reveal a continuum growth law. An intermediate scaling window can therefore be the useful regime. Comparing only two arbitrarily chosen radii can disguise both finite-size and resolution effects.
Fractional scaling
A self-similar construction can have a noninteger scaling exponent. In the standard Koch construction, each segment is replaced by four segments one third as long. The self-similarity relation 4 = 3ᴰ gives D = log(4)/log(3), approximately 1.262.
This exponent is different from the two-dimensional plane used to draw the curve. It also does not mean that each point needs 1.262 coordinate entries. The quantity describes how the ideal construction reproduces its detail under a change of scale.
Mandelbrot's 1967 paper on the coast of Britain brought scale-dependent measurement and fractional dimension into a discussion of irregular geographical curves. A physical coastline has finite observational resolution; an ideal mathematical scaling construction should not silently be extended to every physical scale.
Compare like with like
A model can also use a dynamical probe, such as diffusion and its return probability, to define a dimensional diagnostic. Such a probe depends on the dynamics and on how its time parameter is assigned. A change in the probe need not be a change in the underlying set of sites.
For a claim of emergent physical space, ask which dimensional quantity was calculated, which scale range supports it and what independently connects it to measurement. A rank-three map and a count exponent near three may be compatible without one proving the other.
The SMONAD geometry guides place these questions alongside source relations, locality and clocks. A useful next step is to compute more than one diagnostic within a single explicit construction and investigate the conditions under which they agree.
Sources and further reading
- Mandelbrot, B. (1967), How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, Science 156, 636–638. Published paper.
- Spatial symmetry; topology before measurement; Duality of Time: book contents.